Rome and Carthage fought in several wars that were ongoing for years and years.

These wars between the two powers were called the Punic Wars. The First Punic War lasted for 20 years. Neither side won. Carthage could not beat Rome's army, but Rome could not beat Carthage's Navy. This war and the others caused many deaths and lots of destruction.

During the second war, Hannibal a great Carthaginian general, used a very clever trick in a sea battle. He would win many battles (land and sea) using new and different ideas. But once in 190 BC while fighting at sea, he had his men throw live venomous snakes on to the enemies' ship. The snakes frightened the enemy (Rome's ally, King Eumenes II of Pergamon) so badly they surrendered.

Next, Carthage wanted to conquer all of Iberia and Hannibal did just that. Some cities in Iberia were allies of Rome. When Hannibal attacked one of these cities, Rome had an excuse to declared war (Third Punic War) on Carthage. Rome would win in the end and Carthage was torn down; all existence of Carthage was then completely eliminated. People were sold into slavery. The land was covered with salt and growing food was made impossible.

More Info: en.wikipedia.org